Random Stinespring superchannel: converting channel queries into dilation isometry queries

TL;DR

This paper introduces the random Stinespring superchannel, a method to convert channel queries into isometry queries, leading to optimal quantum channel learning algorithms with proven query complexity bounds.

Contribution

The paper presents a novel superchannel framework that reduces quantum channel learning to isometry learning, and proves the optimality of existing channel tomography algorithms.

Findings

Quantum channel learning reduces to isometry learning.

Optimal query complexity for channel learning established as Θ(d_A d_B r).

Existing algorithms are proven to be optimal, removing previous logarithmic gaps.

Abstract

The recently introduced random purification channel, which converts $n$ copies of an arbitrary mixed quantum state into $n$ copies of the same uniformly random purification, has emerged as a powerful tool in quantum information theory. Motivated by this development, we introduce a channel-level analogue, which we call the random Stinespring superchannel. This consists in a procedure to transform $n$ parallel queries of an arbitrary quantum channel into $n$ parallel queries of the same uniformly random Stinespring isometry, via universal encoding and decoding operations that are efficiently implementable. When the channel is promised to have Choi rank at most $r$, the procedure can be tailored to yield a Stinespring environment of dimension $r$. As a consequence, quantum channel learning reduces to isometry learning, yielding a simple channel learning algorithm, based on existing isometry learning protocols, that matches the performance of the two recently proposed channel tomography algorithms. Complementarily, whereas the optimality of these algorithms had previously been established only up to a logarithmic factor in the dimension, we close this gap by removing this logarithmic factor from the lower bound. Taken together, our results fully establish the optimality of these recently introduced channel learning algorithms, showing that the optimal query complexity of learning a quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ is $\Theta(d_A d_B r)$.